Construction of the log-convex minorant of a sequence { M α } α ∈ N 0 d $\lbrace M_\alpha \rbrace _{\alpha \in \mathbb {N}_0^d}$

We give a simple construction of the log-convex minorant of a sequence ${\left\{M_{\alpha }\right\}}_{\alpha \in N_0^d}$ and consequently extend to the $d$-dimensional case the well-known formula that relates a log-convex sequence ${\left\{M_p\right\}}_{p\in N_0}$ to its associated function ${\omega }_M$, that is, $M_p={sup}_{t>0}{t}^p\exp (-{\omega }_M\left(t\right))$. We show that in the more dimensional anisotropic case the classical log-convex condition $M_{\alpha }^2\le M_{\alpha -e_j}{M}_{\alpha +e_j}$ is not sufficient: convexity as a function of more variables is needed (not only coordinate-wise). We finally obtain some applications to the inclusion of spaces of rapidly decreasing ultradifferentiable functions in the matrix weighted setting.